Notes

It is often convenient to define quantities for which only differences or derivatives matter. In classical physics an example is electric potential, which can be shifted by any constant amount without affecting voltage differences or the electric field given by its gradient. In the mid-1800s the idea emerged of a vector potential whose curl gives the magnetic field, and it was soon recognized—notably by James Clerk Maxwell—that any function whose curl vanishes (and that can therefore normally be written as a gradient) could be added to the vector potential without affecting the magnetic field. By the end of the 1800s the general conditions on electromagnetic potentials for invariance of fields were known, though were not thought particularly significant. In 1918 Hermann Weyl tried to reproduce electromagnetism by adding the notion of an arbitrary scale or gauge to the metric of general relativity (see page 1028)—and noted the "gauge invariance" of his theory under simultaneous transformation of electromagnetic potentials and multiplication of the metric by a position-dependent factor. Following the introduction of the Schrödinger equation in quantum mechanics in 1926 it was almost immediately noticed that the equations for a charged particle in an electromagnetic field were invariant under gauge transformations in which the wave function was multiplied by a position-dependent phase factor. The idea then arose that perhaps some kind of gauge invariance could also be used as the basis for formulating theories of forces other than electromagnetism. And after a few earlier attempts, Yang–Mills theories were introduced in 1954 by extending the notion of a phase factor to an element of an arbitrary non-Abelian group. In the 1970s the Standard Model then emerged, based entirely on such theories. In mathematical terms, gauge theories can be viewed as describing fiber bundles in which connections between values of group elements in fibers at neighboring spacetime points are specified by gauge potentials—and curvatures correspond to gauge fields. (General relativity is in effect a special case in which the group elements are themselves related to spacetime coordinates.)